Intricacies of Addition and Multiplication (4)

We have seen in the last entry how both the quantitative and qualitative nature of the natural numbers is directly related to the operations of addition and multiplication respectively (which are complementary with each other).

Thus in Type 1 terms (where the base is defined in quantitative terms)

by addition,

1 + 1 = 2 (i.e. 1¹ + 1¹ = 2¹).

Then in Type 2 terms (where the dimension is defined in qualitative terms)

by multiplication,

1 * 1 = 2 (i.e. 1¹ * 1¹ = 1²).

Thus we have switched from the quantitative notion of “2” as base number to the qualitative notion of “2” (or twoness)  as dimensional number in this fashion.

Thus with respect to the base (representing specific objects), the quantitative notion of 2 corresponds to cardinal interpretation

So 2 in this context arises from the recognition of homogeneous independent objects (without qualitative distinction)

Then with respect to the dimension (representing general frameworks for objects) the qualitative notion of 2 corresponds directly with ordinal interpretation.

So 2 (as twoness) in this context arises from the recognition of qualitatively distinct  1^st and 2^nd dimensional frameworks (which requires seeing both as qualitatively interdependent with each other). However  as we will indirectly demonstrate later 2 = 1^st + 2^nd lacks any quantitative distinction.

In this way we can see how addition and multiplication are directly related to both the cardinal and ordinal interpretation of the natural numbers respectively.

However Just like the left and right turns at a crossroads are reversed when we approach it from the opposite direction, likewise when we switch the frame of reference (with respect to both quantitative and qualitative) a complementary reverse interpretation results.

So what is addition from a Type 1 perspective, is multiplication from a Type 2 (and vice versa). And this equally applies to both quantitative and qualitative interpretations of base and dimensional values.

So we can equally start with the base number defined as qualitative and the dimensional number as quantitative respectively.

Now addition with respect to the Type 2 aspect implies the quantitative aspect of this dimensional number.

Thus 1 + 1 = 2 (i.e. ₁1 + 1  = ₁2).

Here number representing dimension carries the standard cardinal meaning where 2 = two dimensions.

Then in complementary fashion, multiplication with respect to The Type 1 aspect implies the qualitative aspect the application of this base number.

 

So 1 * 1 = 2 i.e. (₁1 + ₁1   = ₂1).

To distinguish the switch in the meaning (quantitative and qualitative) that numbers now possess, I have likewise reversed the notation, so that base numbers are now represented with subscripts and dimensions as standard size (just as formally, base numbers were represented by normal size and dimensions with superscripts respectively).

Though the meaning associated with the mathematical representation of addition and multiplication is difficult to intuitively grasp (due to the standard identification of number with merely quantitative values) it can be expressed quite simply in psychological terms.

In other words, number perceptions and concepts continually interact in a dynamic manner, whereby both rational (analytic) and intuitive (holistic) aspects are involved.

Through this dynamic interactive process, we are thereby enabled to distinguish the natural numbers in both cardinal and ordinal terms ,where they can represent both (specific) objects and (general) dimensions respectively.

So for example, we are thereby enabled to appreciate 3 as a cardinal number (applying to specific objects); we are also enabled to appreciate 3 in cardinal terms as applying more generally to dimensions i.e. 3 dimensions.

Equally we are enabled to appreciate 3 in ordinal terms with respect to specific objects (as 1^st, 2^nd and 3^rd) and likewise with respect to more generalised dimensions (again as 1^st, 2^nd and 3^rd).

The crucial point to recognise that this crucial capacity - whereby we are enabled to keep switching from cardinal to ordinal (and ordinal to cardinal meaning) - is directly related to the operations of addition and multiplication (that likewise behave in a dynamic interactive manner).

However as long as we attempt to interpret number in a merely quantitative manner, statements regarding the true dynamic nature of addition and multiplication can carry no resonance.

Intricacies of Addition and Multiplication (3)

There are obvious parallels as between the dynamic approach to number and quantum mechanics.

Indeed, I am confident that it will be ultimately understood that this dynamic approach to number serves as the starting basis for quantum mechanical understanding and in that sense is more fundamental. So properly understood, quantum mechanics is rooted in the true dynamic nature of number.

As is well-known all sub-atomic particles manifest themselves in a complementary fashion as both particles and waves.
Likewise, as we have seen, all numbers manifest themselves in a complementary manner through both quantitative (analytic) and qualitative (holistic) aspects.

It might be initially helpful to identify the particle with the quantitative aspect, and the wave with the qualitative aspect respectively.

However just as it is recognised in quantum mechanics that the particle has also a wave, and the wave a particle aspect, likewise we have seen, that when reference frames are switched with respect to number, that the quantitative aspect has a qualitative and the qualitative also a quantitative aspect respectively.

Indeed the parallels go further.

Again as is well-known the uncertainty principle apples with respect to the behaviour of sub-atomic particles. So, for instance, one cannot hope to precisely predict the position and momentum of a particle simultaneously. Rather a trade-off is involved whereby greater precision with respect to one aspect entails increasing imprecision with respect to the other.

This is equally true of number behaviour, whereby an uncertainty principle equally applies to the dynamic behaviour of number. So here for example a trade-off is also involved with respect to simultaneous knowledge of both quantitative (analytic) and qualitative (holistic) aspects.

So greater precision with respect to the quantitative aspect thereby implies greater imprecision with respect to its corresponding qualitative aspect.

This is greatly exemplified by the very nature of Conventional Mathematics. So increasing focus on the merely quantitative aspect of number behaviour has become so extreme that the qualitative aspect is not even recognised in formal interpretation.

So the misleading view that numbers have an absolute quantitative identity (without reference to their qualitative nature) has long become entrenched in accepted understanding.

In truth however numbers enjoy a merely relative identity (based on the dynamic interaction of twin complementary aspects of behaviour).

However the very appreciation of this point will require - as I continually repeat - a radical new paradigm of what Mathematics really represents. Again this "conversion" will I believe signal the greatest revolution yet in our intellectual history!


Before returning directly back to the nature of addition and multiplication, I wish to address a key feature of conventional mathematical understanding that is not properly appreciated.

This relates to its 1-dimensional nature (based on the qualitative holistic meaning 1).

So though dimensions (i.e. powers or exponents) other than 1, are of course recognised in a quantitative (analytic) manner, these are all interpreted within the standard 1-dimensional context (in qualitative terms).

As this is so important, I will comment further here on what it entails.

The 1st dimension is unique is that that it is the only dimension where qualitative and quantitative meaning are reduced in an absolute manner.

So, 1 in the Type 1 aspect of the number system, is represented as 1¹.

However, 1 in the Type 2 aspect of the number system, is likewise represented as 1¹ .

So with respect to 1, no distinction can be made as between its quantitative (Type 1) and qualitative (Type 2) interpretations. This is why a linear (i.e. 1-dimensional) rational approach to interpretation entails the reduction of qualitative to quantitative meaning in an absolute manner.

However what is not at all clearly recognised is that every number (other than 1) can equally serve as a valid means of interpretation of mathematical symbols.

The key distinction then in all these other approaches is that a dynamic relative means of appreciation ensues (entailing the interaction of both analytic and holistic aspects).


So the rigid preoccupation with 1-dimensional interpretation in Mathematics has completely blinded us to the existence of unlimited further terrains of potential meaning (where numbers ≠ 1,serve as the holistic means of interpretation).


We can easily illustrate once more the 1-dimensional approach with respect to number interpretation.


For example if we multiply two numbers, say 2 * 3, the resulting answer is given as 6.

However, if represent this expression in geometrical terms, we can quickly appreciate that a qualitative as well as quantitative conversion takes place. So in quantitative terms the result is indeed 6. However this now relates to 2-dimensional (square) rather than 1-dimensional (linear) units.

However in conventional mathematical terms, this qualitative transformation in the nature of the units is simply ignored with the result expressed in a linear (1-dimensional) manner.

Thus from this perspective 2 * 3 = 6 (i.e. 6¹) . So if you want to appreciate the key problem with respect to reconciling multiplication with addition, it is right here!

Therefore whereas addition (in linear terms) leads solely to a quantitative transformation in the nature of the units, multiplication - by contrast - likewise entails a qualitative transformation.


Indeed from the conventional mathematical perspective, multiplication serves as but a short-hand way of representing addition.

So 2 * 3 (from this perspective) = 2 + 2 + 2. Therefore 2 * 3 expresses the fact that we are adding 2 three times!

However this misses the point completely - as is inevitable from the conventional mathematical perspective - that multiplication essentially, relative to addition, entails a qualitative, rather than quantitative transformation.

The deeper philosophical basis of the 1-dimensional approach is the use of single polar frames of reference in the interpretation of mathematical relationships.

I frequently illustrate this with respect to the interpretation of left and right turns at a crossroads.

I one is heading N, a left turn can be unambiguously identified at the crossroads. So here a single polar frame of reference (i.e. the N direction) is used.

Now having passed through the crossroads one switched direction and then travels S, once again a left turn can be unambiguously identified (using this single pole of reference).

So both turns at the crossroads are now unambiguously identified as left (using separate poles of reference). However we know that the turns must be left and right with respect to each other!

This recognition that identification of turns is merely relative, requires the ability to appreciate N and S as complementary opposite poles (which requires simultaneously linking N and S directions).

The key insight enabling such simultaneous recognition is of a holistic intuitive nature and leads to (circular) paradoxical understanding in terms of unambiguous rational appreciation of an analytic nature.

So absolute unambiguous understanding is based on the use just one polar direction as reference frame. And this is the fundamental nature of linear (1-dimensional) interpretation.

However relative paradoxical understanding always entails simultaneous use of more than one polar reference frame (which in the simplest case entails two).  And this likewise is the fundamental nature of circular (higher dimensional) interpretation.

So rather just one polar reference frame for number (i.e. analytic) we have now extended appreciation to include two such directions i.e. analytic and holistic, in dynamic relationship with each other.


However just as with the interpretation of a crossroads we keep alternating as between absolute understanding (where each turn is unambiguously defined) and relative understanding (where both turns have a merely arbitrary relative interpretation depending on context) likewise we must do the same with dynamic mathematical understanding as we continually alternate as between analytic (linear) and holistic (circular) type appreciation.


Having dealt with these important background issues let us return to the central issue of addition and multiplication.

The remarkable fact that now presents itself, is that once we define both Type 1 and Type 2 aspects of the number system, we can simply distinguish the nature of addition from multiplication.

Thus when we interpret the Type 1 aspect from the standard quantitative perspective, the natural numbers can be derived (as base values) through the continual addition of the starting unit.

So  starting with 1¹, 2¹  = 1¹ + 1¹, 3¹ = 1¹ + 1¹ + 1¹, and so on.

Thus addition is here directly associated with the quantitative transformation of number.


However equally when we interpret the Type 2 aspect from the (unrecognised) qualitative perspective, the natural numbers can be derived (as dimensional values) through the continual multiplication of the starting unit.

So again starting with 1¹,  1²  = 1¹ * 1¹, 1³ = 1¹ * 1¹ * 1¹, and so on.

Thus multiplication is here directly associated with the qualitative transformation of number.

Therefore when correctly understood, addition and multiplication represent two fundamental operations that are quantitative (analytic) and qualitative (holistic) with respect to each other.

However this key distinction cannot be appreciated from a conventional mathematical perspective (where the quantitative aspect is solely recognised).

Intricacies of Addition and Multiplication (2)

We saw yesterday that all numbers - indeed all mathematical notions - possess both quantitative (analytic) and qualitative (holistic) aspects, which dynamically interact with each other in complementary fashion.

Therefore from a comprehensive perspective, every number is thereby defined in terms of both a base and dimensional value (where with respect to the expression a^b, a is the base and b the dimensional value respectively.

From this perspective the natural numbers can be defined with respect to both Type 1 and Type 2 aspects respectively.

With the Type 1 aspect, where the base aspect varies (as quantity), the (default) dimensional number is 1; with the Type 2 aspect, by contrast, where the dimensional aspect (as quality) varies, the default base number is 1.

The cardinal notion of number is then directly associated from this perspective with the Type 1 aspect; the ordinal notion is then directly associated with the Type 2.

Crucially therefore both the cardinal notion (implying independence) with respect to number quantities and the ordinal notion (implying, by contrast, interdependence with respect to the qualitative relationship between numbers) represent two distinct types of number appreciation, which are dynamically relative in nature with respect to each other.


However it is equally possible to switch reference frames. Then with respect to the natural numbers of the Type 1 aspect, the base value now varies and is defined in qualitative (holistic) terms with respect to a (default) dimensional value of 1; then, in relative terms, the dimensional aspect varies, and is now defined in quantitative (analytic) terms with respect to a default base value of 1.

In this way, both base and dimensional values take on quantitative and qualitative meanings that dynamically interact with each other in a complementary fashion.

And if you think that this seems somewhat far-fetched, let me remind you that this is what continually takes place with respect to our experience of number!

So from one perspective we are able to recognise objects in natural number fashion with respect to their cardinal (quantitative) characteristics. So if I point to a group of 3 people on the street, such recognition entails quantitative recognition of number. If however I now specifically identify - say in terms of a criterion such as height - 1st, 2nd and 3rd members of that group - this now entails corresponding ordinal recognition of a qualitative nature.

And likewise it is similar with respect to numbers, now serving as dimensions. So the recognition of 3 dimensions (now serving as more general categories within which specif number objects can be defined) implies cardinal recognition of a quantitative kind. However the distinction as between the 1st, 2nd and 3rd dimensions (according to some criterion) implies ordinal recognition of a qualitative kind.

So with respect to actual experience we effortlessly switch as between both quantitative (analytic) and qualitative (holistic) recognition with respect to natural numbers serving as both objects and dimensions respectively.

However when it comes to conventional mathematical understanding, we are offered but a reduced - and ultimately highly distorted - interpretation of the nature of number.

So rather than two aspects - quantitative and qualitative - being explicitly recognised that are of equal importance, the qualitative is effectively reduced to the quantitative  aspect in a very limited manner.

And then rather that the more accurate relative appreciation of number, as the consequent dynamic interaction as between opposite polarities, the inaccurate notion of number as absolutely existing is an abstract manner has now unfortunately become firmly embedded in consciousness. Indeed it will take the greatest revolution yet witnessed in our intellectual history to enable the required conversion in viewpoint to a truly dynamic perspective.

For what I am really addressing here is the urgent need now now for a completely new kind of mathematical appreciation that can in no way can be incorporated as some kind of extension of existing understanding.


It may help to provide further context for this dynamic interpretation of number to now look at the issue more closely again in terms of the psychological dynamics that underlie number experience.

All phenomenal experience - including of course mathematical - is conditioned by fundamental sets of opposite polarities.

The first set relates to external and internal poles. So what we identify as objectively existing strictly has no meaning independent of the internal mental constructs used to interpret such experience.
Thus in truth, experience represents a continual dynamic interaction with respect to objects as external and corresponding mental constructs, which relatively of an internal nature.

Thus all objective understanding necessarily reflects a certain (arbitrary) interpretation with respect to such experience.

So for example the belief that numbers represent absolute objects enjoying an abstract existence in a universal mathematical Heaven, ultimately reflects a distorted interpretation (i.e. that numbers can have an objective existence independent of interpretation).
Therefore once we recognise the necessary interaction of (external) objects with (internal) interpretation, mathematical reality must then be understood in a dynamic relative fashion.


The second set of key polarities relates to the interaction of whole and part, which equally manifests itself in terms of quantitative and qualitative aspects that are general and specific with respect to each other.

It is this second set that is directly relevant to the dynamic interpretation of number here outlined.
In the experiential interaction of polarities, we always necessarily experience these poles - to a degree that can greatly vary  - as both independent from us and yet interdependent with us in experience.

For example to recognise and object as external, we thereby need to experience it as - relatively - independent from us. However it cannot be experienced as completely independent as this would exclude any mental interaction (thereby eliminating the possibility of experience).

Therefore, with respect to our conscious recognition, we experience the object as independent;  however with respect to our unconscious recognition (of which we may not be explicitly aware) both external and internal aspects are recognised as interdependent with us.

Then with respect to mathematical understanding, the independent aspect (of conscious recognition) is directly identified in analytic fashion with rational understanding (strictly linear rational understanding).

However the corresponding interdependent aspect (of unconscious recognition) is directly identified in holistic manner with intuitive appreciation (which indirectly can be given a paradoxical circular rational expression) .

Therefore the recognition that number has dual aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other, equally implies the incorporation of (conscious) reason with unconscious (intuition).

Therefore Mathematics can no longer be identified as merely a conscious rational pursuit. Rather the much greater requirement now exists to successfully reconcile both (conscious) reason with (unconscious) intuition with respect to all mathematical understanding .

Though intuition may indeed be informally recognised as important for creative mathematical discovery, in formal terms it is reduced to reason. Thus the failure to recognise both quantitative (analytic) and qualitative (holistic) aspects of number as clearly distinct, directly parallels the corresponding failure to explicitly recognise (conscious) reason and (unconscious) intuition as distinct with respect to understanding.

The relationship between number as representing base and dimensional objects respectively, corresponds directly in psychological terms with the corresponding relationship as between perceptions and concepts.

In the dynamics of understanding, when a number perception takes place in quantitative, terms, a complementary conceptual recognition of that number takes place in a qualitative manner.

So when the natural number perception is of a cardinal nature, the corresponding conceptual recognition is - relatively - ordinal.

However. equally when the number perception takes place in a qualitative manner, the complementary conceptual recognition is qualitative.

Therefore when the natural number perception is now ordinal in nature, the corresponding conceptual recognition is - relatively - cardinal.

In this manner, through the dynamics of experience we keep switching as between both cardinal and ordinal recognition of natural numbers in base and dimensional number terms.
In other words we can recognise natural numbers as applying to both specific objects and general dimensions in both a cardinal and ordinal manner!

However when a distorted interpretation is imposed on this number process, whereby only the quantitative aspects is recognised (thereby explicitly conforming to linear rational interpretation), it sets severe limits with respect to the nature of interaction than can take place. Remember smooth interaction requires recognition of complementary quantitative and qualitative aspects!

So in conventional mathematical terms, the interaction between opposite polarities (quantitative and qualitative) becomes so rigid that the qualitative aspect is no longer even recognised.
The mistaken belief in a merely quantitative interpretation (corresponding to linear rational interpretation) then prevails.

This has become so ingrained - representing the deep unrecognised shadow of Conventional Mathematics - that I would thereby expect enormous resistance with respect to the appropriate dynamic interpretation of number (that properly concurs with experience) .

Intricacies of Addition and Multiplication (1)

The apparently simple operation of multiplication reveals in a glaring fashion the stark inadequacies of conventional mathematical interpretation. In fact the true nature of multiplication cannot be properly explained from this perspective!

The starting point in rectifying this problem is the clear recognition that - far from being static absolute identities - all numbers are inherently dynamic in nature entailing the interaction of twin complementary aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.

For example the number 2 is a cardinal number with a recognised independent quantitative meaning.

This implies that it is defined in a manner that renders it as without any meaningful qualitative (i.e. relational) context.

If numbers were indeed independent in an absolute manner, then by definition, no means would exist for establishing their interdependence (i.e. relationship) with other numbers!

So what we have in conventional mathematical terms is but a reduced interpretation of number. Thus the critical issue of establishing a relationship with other numbers - which is inherently of a qualitative nature - is effectively ignored!

So what is completely overlooked in formal conventional terms, is that every number - which I am illustrating here in the specific case of 2 - has strictly but a relative meaning with  both quantitative and qualitative aspects that dynamically interact with each other in a complementary (opposite) manner.

So once again the number "2" has a quantitative (analytic) meaning. So for example, if I look out my front window and see two cars in the driveway, I am using "2" in its accepted quantitative sense.

However "2" has an equally important meaning that is of a qualitative (holistic) nature which can be referred to as "twoness" (i.e. the quality of "2").

Now whereas the former aspect relates to the independent nature of number recognition in a quantitative manner, the latter aspect - by contrast - relates to the corresponding interdependent nature of number which thereby provides for number its crucial relationship context.

In this way through the interaction of both quantitative and qualitative aspects, we are enabled to recognise number relationships in relative fashion, entailing both independence (from other numbers as distinct entities) and interdependence (through common relationship with these other numbers) respectively.

Now when we define the component "building blocks" of "2" in a quantitative manner, we do so in a manner that strictly lacks any qualitative distinction. To facilitate further exposition, I will refer to this as the Type 1 aspect of number!

So from the Type 1 perspective, 2 = 1 + 1, Here each unit is defined in a strictly homogeneous manner (where neither can be meaningfully distinguished from each other).

This begs the very fundamental question is how one is enabled to form number recognition of "2", given that its unit components are defined in an independent manner! So once again the all important qualitative aspect,whereby meaningful relationship as between separate units can be established,is entirely missing from conventional mathematical interpretation.

When we look at the latter aspect of number (i.e. Type 2), we arrive at a complementary appreciation of number that is qualitative (and thereby strictly lacking any quantitative distinction).

Then from the Type 2 perspective 2 (now reflecting its qualitative nature as "twoness") enables us to clearly distinguish the unique nature of both its units.

So here, 2 = 1st + 2nd, where each unit is now uniquely defined in an ordinal manner.

.
It is vitally important to realise in this context that both the cardinal and ordinal nature of number are directly identified with the two aspects of number (Type 1 and Type 2) respectively, which are crucially distinct from each other. This once again highlights the significantly reduced accepted interpretation, where both cardinal and ordinal aspects are absolutely treated in a merely (Type 1)) quantitative manner.


We are now ready to make the all important leap, whereby the true distinctive nature of addition and multiplication can be clearly revealed.

Because we are now recognising two aspects to number (that dynamically interact in experiential understanding), all numbers are defined in a twin manner containing both a base and a dimensional aspect.
Furthermore to move from Type 1 to Type 2 recognition we invert both aspects in a complementary manner.

So in Type 1 terms the number "2" is more comprehensively defined as 2¹. So the base  here is 2 and the dimensional aspect 1.

The conventional treatment of natural numbers effectively views numbers solely in Type 1 terms (where they are given an absolute interpretation).

So 1, 2, 3, 4, ... can be more fully defined as 1¹, 2¹, 3¹, 4¹...

However because the reduced quantitative value in each case remains unchanged (when the dimensional number is 1), the implied default dimension (i.e. 1) is omitted altogether.

Furthermore whenever number expressions entail dimensional values (powers or exponents) other than 1, the ultimate value is given in a reduced quantitative manner (defined in terms of 1 as dimension).

So for example, in conventional mathematical terms 2² = 4 (i.e.4¹). So though in geometrical terms, we can easily see that this would represent 4 square (i.e. 2-dimensional) units - rather than 4 linear (1-dimensional) units, this qualitative change in the nature of the units is simply ignored, with the resulting value i.e. 4 given in a merely reduced  quantitative manner (i.e. in 1-dimensional terms).

In Type 2 terms  however the number "2" is defined in a complementary manner as 1² (where base and dimensional aspects are switched).

Thus the dimensional number here now varies (with respect to a fixed base number of 1) directly indicating the true qualitative nature of the number "2". Now once again this relates to the qualitative recognition of 2 as a number group whereby both 1st and 2nd members can be uniquely distinguished in an ordinal fashion.

However from a conventional perspective, the Type 2 aspect of number seems pointless, as the reduced quantitative value of each number = 1 (i.e. 1¹).


Finally in this entry, I wish to highlight the dynamic nature of interpretation that is now required (when we attempt to reconcile both Type 1 and Type 2 aspects with each other.

Expressing it simply, the numbers representing base and dimensional values respectively are always opposite to each other.

Therefore if - as we have seen with the Type 1 aspect - the base number is defined in a quantitative (analytic) manner, the corresponding (default) dimensional number is now - relatively - of a qualitative (holistic) nature.

However if by contrast - as we have seen with the Type 2 aspect - the dimensional number is defined in a qualitative (holistic) manner, the corresponding (default) base number (i.e. 1) is then of a quantitative (analytic) nature.


However, just as in the manner that the directions of left and right turns at a crossroads are reversed when we approach it from an opposite direction, likewise, when the polar frame of reference switches with respect to number the base number can now take on a qualitative (holistic) meaning, while the dimensional number - now relatively - is of a quantitative (analytic) nature.

We will elaborate further in the next entry!

Reflections on Experience (13)

My interest in the Riemann Hypothesis itself reflected a deeper wonder regarding the ultimate nature of the number system.

As by extension the nature of number fundamentally underlies wider mathematical understanding and the intimate underpinning of physics (together with all the other sciences), in this sense nothing indeed could be more important that the nature of number!

And what I finally discovered may seem utterly surprising - and even shocking - to the interested reader.

For ultimately, in truth, a holistic synchronicity (where everything is co-determined in simultaneous fashion) lies at the very heart of the number system.

Central to this appreciation is the realisation that all numbers - indeed all mathematical symbols - possess two distinct aspects of equal importance, that are quantitative (analytic) and qualitative (holistic) with respect to each other.

When seen in this light, what we conventionally accept as Mathematics represents but a limited form of understanding, where in every context the qualitative aspect is reduced in an absolute quantitative type manner. This in effect entails the attempt to view numbers as static independent entities (while ignoring the need to establish their related interdependence with all other numbers).

When however one recognises the equal importance of both quantitative and qualitative aspects, the number system is then appropriately understand in a relative manner (entailing the dynamic interaction of both aspects).   One key consequence of this understanding is that ordinal is seen as utterly distinct from cardinal type interpretation.

Another key consequence of this recognition of two distinct aspects of number (which I commonly refer to as Type 1 and Type 2 respectively) is that parallel sets of zeta zeros exist for both types.

So just as we have the famed Riemann zeros that exist with respect to the Type 1 (which I refer to as the Zeta 1 zeros) equally we have a largely unrecognised set that likewise exist for the Type 2 (which I refer to as the Zeta 2 zeros).

When seen from this new more comprehensive perspective, the key significance of both sets of zeros is that vitally, they enable seamless conversion to take place as between both aspects of the number system (Type 1 and Type 2).

So the Zeta 1 zeros enable conversion from Type 1 to Type 2 format; in complementary fashion, the Zeta 2 zeros enable conversion from Type 2 to Type 1 format.

Thus the zeta zeros (both sets) are fundamentally important in enabling the consistent interplay of number notions in both cardinal and ordinal fashion.

The relationship of the primes and natural numbers is equally important in this understanding, for it is through the two-way relationship of the primes to the natural numbers (and the natural numbers to the primes) in cardinal and ordinal manner, that both the quantitative and qualitative aspects of number are transmitted.

So far from number being misleadingly viewed an independent abstract entity in an absolute objective manner, number is now seen as deeply embedded in all phenomenal processes (in both physical and psychological terms).


We are slowly coming to accept that the physical world fundamentally operates in a very different manner from what we accustomed to believe.

Thus at the sub-atomic level of quantum processes, the very notion of independent particles (such as electrons) has no strict meaning.

And the non-local effect of particles, which really represent the unrecognised holistic behaviour of such phenomena, has been amply demonstrated (without however its startling philosophical implications being seriously addressed).

However what is now even more startling is that all these quantum type effects intimately apply likewise to the nature of number.

Indeed it is even more shocking in a sense in that all this strange quantum behaviour of matter is itself ultimately rooted in the true nature of number!

What is not yet sufficiently realised is that our common intuitions about physical reality are themselves deeply rooted in our limited mathematical notions of number (where they are given an absolute type identity).

 So the revolution in physical understanding - implied by recent quantum developments - cannot properly take place until a deeper revolution takes place with respect to our understanding of number.

And when this eventually happens, it will signal by far the greatest revolution yet in our intellectual history.

When properly understood, in a more comprehensive dynamic manner, it then becomes immediately obvious why the Riemann Hypothesis can neither be proved (nor disproved) in conventional mathematical terms.

The Riemann Hypothesis in truth is pointing to the condition required for the prior consistency of both the quantitative (analytic) and qualitative (holistic) aspects of number. However as the qualitative aspect is simply reduced to the quantitative in conventional terms, this means that the truth of the Riemann Hypothesis is already implied by conventional mathematical axioms.

So the Riemann Hypothesis transcends conventional mathematical notions (while also being prior to all such understanding).

Thus the truth of the Riemann Hypothesis in the end represents an act of faith in the subsequent consistency of the whole mathematical enterprise!

Reflections on Experience (12)

When I initially decided to tackle the Riemann Hypothesis, one key issue intrigued me with respect to negative values of the Riemann Zeta Function.

For example when s = – 2,

ζ(– 2) = 0.


However this seemed to me very puzzling as,

 ζ(– 2) = 1/1 ^{– 2} + 1/2 ^{– 2} + 1/3 ^{– 2} + .........

= 1 + 4 + 9  + ......


Thus in terms of conventional analytic interpretation, this series diverges to infinity.

Yet, paradoxically in terms of the Riemann zeta function its value is given as 0!

Though typically, negative values of s are derived through the accepted  process of analytic continuation, I could find no convincing mathematical explanation given anywhere as to why such a quantitative series can lead to two utterly distinct values.

Eventually I managed to come up with an explanation, which however required re-interpretation of the Riemann Zeta Function in a radical new manner.

What I concluded was that in effect all mathematical relationships can be given two complementary interpretations that are analytic and holistic with respect to each other.

Then with respect to the Riemann zeta function, where the value of s is positive and > 1, a quantitative value arises that concurs directly with standard analytic interpretation.

However where the value of s is negative and < 1, a quantitative value arises that, in complementary fashion, concurs directly with holistic interpretation.

Thus the significance of the Riemann zeta function where for any value of ζ(s), a corresponding value can be given for ζ(1 –  s), arises from the fact that it directly relates quantitative values, with an analytic interpretation on the RHS of the real axis, with corresponding values with a complementary holistic interpretation on the LHS.

It is here that the many years work I had devoted to holistic mathematical interpretation paid great dividends.

So I had already holistically defined the dimension 2 in a circular manner as representing the complementarity of opposite poles ( + 1 and – 1).

However this holistic meaning of 2 (as dimension) in turn can be given positive and negative values (i.e. as + 2 and – 2 respectively).

When + 2 is used, this signifies the (indirect) rational attempt to translate the paradoxical notion of complementary opposites (i.e. that simultaneously are both positive and negative with respect to each other)

When – 2 is used this signifies however the direct intuitive realisation of this relationship (which intrinsically is of a qualitative nature). So such intuitive realisation is thereby literally nothing in quantitative terms (which implies the holistic meaning of 0).

I was helped towards this realisation through a deep immersion in the works of St. John of the Cross who deals intensively with the process of "passive" purgation" in the spiritual life.

The goal of such purgation is to cleanse the mind of indirect conscious attachment to symbols that intrinsically are designed to serve a nondual spiritual purpose.

And the goal of such cleansing - as put so starkly by St. John - is to achieve "nada" (i.e. a state of nothingness or spiritual emptiness).

Thus in the context of mathematical understanding, such contemplative cleansing would be designed to reveal the pure holistic meaning of symbols (that are directly understood in an intuitive manner).

Thus, when correctly understood, all mathematical symbols have both rational (analytic) and intuitive (holistic) meanings with - in dynamic experiential terms - a dynamic interaction continually taking place as between both aspects. However in conventional mathematical terms the intuitive (holistic) aspect is reduced to rational (analytic) interpretation in an absolute fashion!

So when one holistically interprets the value of the Riemann zeta function that corresponds to – 2, this implies appreciation that relates to the direct intuitive realisation of the complementarity of two opposite poles (which is 0 in quantitative terms).


Thus through Riemann's functional equation, we can directly relate analytic quantitative values on the RHS of the real axis (where s > 1) with corresponding holistic quantitative values on the LHS (corresponding to 1 – s).

Crucially in the critical region (where 0 < s < 1), values arising share both analytic (quantitative) and holistic (qualitative) characteristics.

The significance therefore of the condition (to which the Riemann Hypothesis relates) that s = .5, is that through the Riemann function, this now signifies the vital condition where both analytic (quantitative) and holistic (qualitative) values coincide.

The additional requirement that the value of the zeta function = 0 (for all values of s) then directly relates to the non-trivial zeros of the function.

Thus meaningful interpretation of the Riemann zeta function (and associated Riemann Hypothesis)  requires a dynamic interactive appreciation entailing the two-way complementary relationship of numerical values (that can be given both analytic and holistic meanings).

From this dynamic perspective the non-trivial zeros represents those points (represented on the imaginary axis trough .5) where both the analytic (quantitative) and holistic (qualitative) interpretation of mathematical values directly coincide.  
Alternatively this could be expressed as those points where both randomness and order dynamically coincide with respect to operation of the number system.